About the project

Introduction to Open Data Science course, my github repository: https://github.com/peempe/IODS-project

Regression and model validation

Learning2014 dataset includes information about students who attended the basic statistics course at the end of year 2014. The dataset includes 7 variables and 166 observations. The variables are gender, age, attitude towards statistics, points in the final exam and variables addressing deep, surface, and strategic learning.

More information about the dataset can be found here.

First, we need to get the dataset to R. I had some problems with my Mac (even though I double checked it like thousand times in the “Data wrangling” part), so I decided to download it via link.

setwd("~/Documents/GitHub/IODS-project/data")
learning2014 <- read.table("http://s3.amazonaws.com/assets.datacamp.com/production/course_2218/datasets/learning2014.txt",
                           sep = ",", header = T)

Overview of the data

Second, we need to have an access to ggplot2 and GGally-packages. Remember to install them first!

library(ggplot2)
library(GGally)

Next, let’s see how the data looks like.

plot1 <- ggpairs(learning2014, mapping = aes(col = gender, alpha = 0.3), lower = list(combo = wrap("facethist", bins = 20)))
plot1

Here are some main points about the output:

  • Most of the participants are women in their twenties.

  • The highest correlation is between attitude and point, addressing that a good attitude towards statistics and/or the course is associated with higher points in the final exam.

  • The lowest correlation is between deep learning strategy and point, indicating that they have nothing to do with each others.

Regression model

Now we could examine a bit do gender, age, and attitude predict points in the final exam.

points_model <- lm(points ~ gender + age + attitude, data = learning2014)
summary(points_model)
## 
## Call:
## lm(formula = points ~ gender + age + attitude, data = learning2014)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -17.4590  -3.3221   0.2186   4.0247  10.4632 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 13.42910    2.29043   5.863 2.48e-08 ***
## genderM     -0.33054    0.91934  -0.360    0.720    
## age         -0.07586    0.05367  -1.414    0.159    
## attitude     3.60657    0.59322   6.080 8.34e-09 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 5.315 on 162 degrees of freedom
## Multiple R-squared:  0.2018, Adjusted R-squared:  0.187 
## F-statistic: 13.65 on 3 and 162 DF,  p-value: 5.536e-08

In the model above, only attitude is statistically significant variable with p-value < .001. The model shows that better attitude towards statistics/course precits higher points in the exam. Because gender and age are not significant variables, we can remove them from the model and run the model again only with attitude-variable.

points_att <- lm(points ~ attitude, data = learning2014)
summary(points_att)
## 
## Call:
## lm(formula = points ~ attitude, data = learning2014)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -16.9763  -3.2119   0.4339   4.1534  10.6645 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  11.6372     1.8303   6.358 1.95e-09 ***
## attitude      3.5255     0.5674   6.214 4.12e-09 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 5.32 on 164 degrees of freedom
## Multiple R-squared:  0.1906, Adjusted R-squared:  0.1856 
## F-statistic: 38.61 on 1 and 164 DF,  p-value: 4.119e-09

The attitude is still highly significant and it explains 19 % of the value of points (R-square). Note, that the R-square value did not change much when the two other variables were removed from the model.

Diagnostic plots

Last, after fitting the model and intepreting the results, we should check the diagnostic plots. The following commands do the trick:

points_att <- lm(points ~ attitude, data = learning2014)
plot(points_att, which = c(1, 2, 5), par(mfrow = c(2,2)))

All plots shows that the residuals are nicely spread on both sides of 0 (Residuals vs Fitted values), follow the line well and are (quite) normally distributed (Normal QQ-plot), and the few outliers do not influence the analysis too much (Residuals vs Leverage). The plots indicate that our model fit well to the data and the association between attitude and points is linear (as it should be when using linear regression).

Logistic regression

Reading the data

alc <- read.table("http://s3.amazonaws.com/assets.datacamp.com/production/course_2218/datasets/alc.txt", 
                  sep = ",", header = T)
View(alc)
colnames(alc)
##  [1] "school"     "sex"        "age"        "address"    "famsize"   
##  [6] "Pstatus"    "Medu"       "Fedu"       "Mjob"       "Fjob"      
## [11] "reason"     "nursery"    "internet"   "guardian"   "traveltime"
## [16] "studytime"  "failures"   "schoolsup"  "famsup"     "paid"      
## [21] "activities" "higher"     "romantic"   "famrel"     "freetime"  
## [26] "goout"      "Dalc"       "Walc"       "health"     "absences"  
## [31] "G1"         "G2"         "G3"         "alc_use"    "high_use"

The dataset is jointed from two different datasets and it includes information about students’ alcohol consumpiton. The infromation is gathered from Portuguese and Math classes from two Portuguese schools

More information about the dataset can be found here.

The research plan and hypotheses

The aim of this week is to analyze the relationship of high/low alcohol consumption (AC) and following variables: mother’s education (Medu), quality of family realtionships (famrel), current health status (health), and number of school absences (absences).

My hypotheses are: 1. Mother’s low education is associated with higher AC. 2. Poor quality of family relationships is associated with higher AC. 3. Poor health status is associated with higher AC. 4. Higher number of school absences is associated with higher AC.

Preliminary analyses

library(ggplot2)

Mother’s education

Medutable <- table(alc$Medu,alc$high_use)
Medutable
##    
##     FALSE TRUE
##   0     1    2
##   1    33   18
##   2    80   18
##   3    59   36
##   4    97   38

Quality of family relationships

FamRtable <- table(alc$famrel,alc$high_use)
FamRtable
##    
##     FALSE TRUE
##   1     7    2
##   2     9    9
##   3    40   26
##   4   131   52
##   5    83   23

Current health status

Healthtable <- table(alc$health,alc$high_use)
Healthtable
##    
##     FALSE TRUE
##   1    35   11
##   2    29   14
##   3    63   20
##   4    47   17
##   5    96   50

Number of school absences

g1 <- ggplot(alc, aes(x = high_use, y = absences, col = sex))
g1 + geom_boxplot() + ggtitle("School absences by alcohol consumption")

Logistic regression

library(magrittr)

m <- glm(high_use ~ Medu + famrel + health + absences, data = alc, family = "binomial")
summary(m)
## 
## Call:
## glm(formula = high_use ~ Medu + famrel + health + absences, family = "binomial", 
##     data = alc)
## 
## Deviance Residuals: 
##     Min       1Q   Median       3Q      Max  
## -2.8333  -0.8113  -0.6894   1.1883   1.9684  
## 
## Coefficients:
##             Estimate Std. Error z value Pr(>|z|)    
## (Intercept) -0.50040    0.63728  -0.785 0.432326    
## Medu        -0.05114    0.10800  -0.474 0.635825    
## famrel      -0.29326    0.12388  -2.367 0.017918 *  
## health       0.14367    0.08676   1.656 0.097742 .  
## absences     0.06806    0.01799   3.783 0.000155 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for binomial family taken to be 1)
## 
##     Null deviance: 462.21  on 381  degrees of freedom
## Residual deviance: 435.72  on 377  degrees of freedom
## AIC: 445.72
## 
## Number of Fisher Scoring iterations: 4
# computing odds ratios (OR)
OR <- coef(m) %>% exp

# computing confidence intervals (CI)
CI <- confint(m) %>% exp
## Waiting for profiling to be done...
# printing out the odds ratios with their confidence intervals
cbind(OR, CI)
##                    OR     2.5 %    97.5 %
## (Intercept) 0.6062858 0.1712953 2.0992320
## Medu        0.9501429 0.7689690 1.1754068
## famrel      0.7458269 0.5839293 0.9507828
## health      1.1545005 0.9762890 1.3729075
## absences    1.0704246 1.0353696 1.1107908

Predictive power of the model

library(dplyr); library(ggplot2)
## 
## Attaching package: 'dplyr'
## The following object is masked from 'package:GGally':
## 
##     nasa
## The following objects are masked from 'package:stats':
## 
##     filter, lag
## The following objects are masked from 'package:base':
## 
##     intersect, setdiff, setequal, union
# predicting the probability of high_use
probabilities <- predict(m, type = "response")

# adding the predicted probabilities to 'alc'
alc <- mutate(alc, probability = probabilities)

# using the probabilities to make a prediction of high_use
alc <- mutate(alc, prediction = probability > 0.5)

# seeing the last ten original classes, predicted probabilities, and class predictions
select(alc, failures, absences, sex, high_use, probability, prediction) %>% tail(10)
##     failures absences sex high_use probability prediction
## 373        1        0   M    FALSE   0.2677134      FALSE
## 374        1       14   M     TRUE   0.3127047      FALSE
## 375        0        2   F    FALSE   0.2045304      FALSE
## 376        0        7   F    FALSE   0.3355262      FALSE
## 377        1        0   F    FALSE   0.2057626      FALSE
## 378        0        0   F    FALSE   0.1566707      FALSE
## 379        1        0   F    FALSE   0.4684247      FALSE
## 380        1        0   F    FALSE   0.4684247      FALSE
## 381        0        3   M     TRUE   0.3210711      FALSE
## 382        0        0   M     TRUE   0.2481431      FALSE
# tabulating the target variable versus the predictions
table(high_use = alc$high_use, prediction = alc$prediction)
##         prediction
## high_use FALSE TRUE
##    FALSE   258   12
##    TRUE     96   16
# initialize a plot of 'high_use' versus 'probability' in 'alc'
g <- ggplot(alc, aes(x = probability, y = high_use, col = prediction))

# define the geom as points and draw the plot
g + geom_point()

# tabulate the target variable versus the predictions
table(high_use = alc$high_use, prediction = alc$prediction) %>% prop.table() %>% addmargins()
##         prediction
## high_use      FALSE       TRUE        Sum
##    FALSE 0.67539267 0.03141361 0.70680628
##    TRUE  0.25130890 0.04188482 0.29319372
##    Sum   0.92670157 0.07329843 1.00000000
# define a loss function (mean prediction error)
loss_func <- function(class, prob) {
  n_wrong <- abs(class - prob) > 0.5
  mean(n_wrong)
}

# call loss_func to compute the average number of wrong predictions in the (training) data
loss_func(class = alc$high_use, prob = alc$probability)
## [1] 0.2827225

10-fold cross-validation

# 10-fold cross-validation
library(boot)
cv <- cv.glm(data = alc, cost = loss_func, glmfit = m, K = 10)

# average number of wrong predictions in the cross validation
cv$delta[1]
## [1] 0.2827225

Sorry, I didn’t have time to finish the exercise.

Clustering and classification

Overview of the data

The dataset is an example dataset in package MASS. It includes information of housing values in suburbs of Boston

More information about the dataset can be found here.

library(magrittr)
library(MASS)
## 
## Attaching package: 'MASS'
## The following object is masked from 'package:dplyr':
## 
##     select
# Load the data
data("Boston")

# Explore the dataset
str(Boston) # 506 rows/obervations, 14 columns/variables
## 'data.frame':    506 obs. of  14 variables:
##  $ crim   : num  0.00632 0.02731 0.02729 0.03237 0.06905 ...
##  $ zn     : num  18 0 0 0 0 0 12.5 12.5 12.5 12.5 ...
##  $ indus  : num  2.31 7.07 7.07 2.18 2.18 2.18 7.87 7.87 7.87 7.87 ...
##  $ chas   : int  0 0 0 0 0 0 0 0 0 0 ...
##  $ nox    : num  0.538 0.469 0.469 0.458 0.458 0.458 0.524 0.524 0.524 0.524 ...
##  $ rm     : num  6.58 6.42 7.18 7 7.15 ...
##  $ age    : num  65.2 78.9 61.1 45.8 54.2 58.7 66.6 96.1 100 85.9 ...
##  $ dis    : num  4.09 4.97 4.97 6.06 6.06 ...
##  $ rad    : int  1 2 2 3 3 3 5 5 5 5 ...
##  $ tax    : num  296 242 242 222 222 222 311 311 311 311 ...
##  $ ptratio: num  15.3 17.8 17.8 18.7 18.7 18.7 15.2 15.2 15.2 15.2 ...
##  $ black  : num  397 397 393 395 397 ...
##  $ lstat  : num  4.98 9.14 4.03 2.94 5.33 ...
##  $ medv   : num  24 21.6 34.7 33.4 36.2 28.7 22.9 27.1 16.5 18.9 ...
summary(Boston)
##       crim                zn             indus            chas        
##  Min.   : 0.00632   Min.   :  0.00   Min.   : 0.46   Min.   :0.00000  
##  1st Qu.: 0.08204   1st Qu.:  0.00   1st Qu.: 5.19   1st Qu.:0.00000  
##  Median : 0.25651   Median :  0.00   Median : 9.69   Median :0.00000  
##  Mean   : 3.61352   Mean   : 11.36   Mean   :11.14   Mean   :0.06917  
##  3rd Qu.: 3.67708   3rd Qu.: 12.50   3rd Qu.:18.10   3rd Qu.:0.00000  
##  Max.   :88.97620   Max.   :100.00   Max.   :27.74   Max.   :1.00000  
##       nox               rm             age              dis        
##  Min.   :0.3850   Min.   :3.561   Min.   :  2.90   Min.   : 1.130  
##  1st Qu.:0.4490   1st Qu.:5.886   1st Qu.: 45.02   1st Qu.: 2.100  
##  Median :0.5380   Median :6.208   Median : 77.50   Median : 3.207  
##  Mean   :0.5547   Mean   :6.285   Mean   : 68.57   Mean   : 3.795  
##  3rd Qu.:0.6240   3rd Qu.:6.623   3rd Qu.: 94.08   3rd Qu.: 5.188  
##  Max.   :0.8710   Max.   :8.780   Max.   :100.00   Max.   :12.127  
##       rad              tax           ptratio          black       
##  Min.   : 1.000   Min.   :187.0   Min.   :12.60   Min.   :  0.32  
##  1st Qu.: 4.000   1st Qu.:279.0   1st Qu.:17.40   1st Qu.:375.38  
##  Median : 5.000   Median :330.0   Median :19.05   Median :391.44  
##  Mean   : 9.549   Mean   :408.2   Mean   :18.46   Mean   :356.67  
##  3rd Qu.:24.000   3rd Qu.:666.0   3rd Qu.:20.20   3rd Qu.:396.23  
##  Max.   :24.000   Max.   :711.0   Max.   :22.00   Max.   :396.90  
##      lstat            medv      
##  Min.   : 1.73   Min.   : 5.00  
##  1st Qu.: 6.95   1st Qu.:17.02  
##  Median :11.36   Median :21.20  
##  Mean   :12.65   Mean   :22.53  
##  3rd Qu.:16.95   3rd Qu.:25.00  
##  Max.   :37.97   Max.   :50.00
# Calculate the correlation matrix and round it
cor_matrix<-cor(Boston) %>% round(digits = 2)

# Print the correlation matrix
cor_matrix
##          crim    zn indus  chas   nox    rm   age   dis   rad   tax
## crim     1.00 -0.20  0.41 -0.06  0.42 -0.22  0.35 -0.38  0.63  0.58
## zn      -0.20  1.00 -0.53 -0.04 -0.52  0.31 -0.57  0.66 -0.31 -0.31
## indus    0.41 -0.53  1.00  0.06  0.76 -0.39  0.64 -0.71  0.60  0.72
## chas    -0.06 -0.04  0.06  1.00  0.09  0.09  0.09 -0.10 -0.01 -0.04
## nox      0.42 -0.52  0.76  0.09  1.00 -0.30  0.73 -0.77  0.61  0.67
## rm      -0.22  0.31 -0.39  0.09 -0.30  1.00 -0.24  0.21 -0.21 -0.29
## age      0.35 -0.57  0.64  0.09  0.73 -0.24  1.00 -0.75  0.46  0.51
## dis     -0.38  0.66 -0.71 -0.10 -0.77  0.21 -0.75  1.00 -0.49 -0.53
## rad      0.63 -0.31  0.60 -0.01  0.61 -0.21  0.46 -0.49  1.00  0.91
## tax      0.58 -0.31  0.72 -0.04  0.67 -0.29  0.51 -0.53  0.91  1.00
## ptratio  0.29 -0.39  0.38 -0.12  0.19 -0.36  0.26 -0.23  0.46  0.46
## black   -0.39  0.18 -0.36  0.05 -0.38  0.13 -0.27  0.29 -0.44 -0.44
## lstat    0.46 -0.41  0.60 -0.05  0.59 -0.61  0.60 -0.50  0.49  0.54
## medv    -0.39  0.36 -0.48  0.18 -0.43  0.70 -0.38  0.25 -0.38 -0.47
##         ptratio black lstat  medv
## crim       0.29 -0.39  0.46 -0.39
## zn        -0.39  0.18 -0.41  0.36
## indus      0.38 -0.36  0.60 -0.48
## chas      -0.12  0.05 -0.05  0.18
## nox        0.19 -0.38  0.59 -0.43
## rm        -0.36  0.13 -0.61  0.70
## age        0.26 -0.27  0.60 -0.38
## dis       -0.23  0.29 -0.50  0.25
## rad        0.46 -0.44  0.49 -0.38
## tax        0.46 -0.44  0.54 -0.47
## ptratio    1.00 -0.18  0.37 -0.51
## black     -0.18  1.00 -0.37  0.33
## lstat      0.37 -0.37  1.00 -0.74
## medv      -0.51  0.33 -0.74  1.00
# Visualize the correlation matrix
library(corrplot)
## corrplot 0.84 loaded
corrplot(cor_matrix, method="circle", type = "upper", cl.pos = "b", tl.pos = "d", tl.cex = 0.6)

The correlation matrix gives us a nice overview of the relationships between the variables. By using colors to indicate the direction and strength of the correlation, we can see the relevant information with quick look. The matrix shows that the strongest negative correlation is between variables “rad” and “tax”. This indicates that higher index of accessibility to radial highways is associated with lower full-value property-tax rate per $10,000. The strong positive correlation can be found e.g. between variables “age” and “dis” addressing that higher proportion of owner-occupied units built prior to 1940 is associated with higher weighted mean of distances to five Boston employment centres.

Scaling the dataset

# Center and standardize variables
boston_scaled <- scale(Boston)

# Summaries of the scaled variables
summary(boston_scaled)
##       crim                 zn               indus        
##  Min.   :-0.419367   Min.   :-0.48724   Min.   :-1.5563  
##  1st Qu.:-0.410563   1st Qu.:-0.48724   1st Qu.:-0.8668  
##  Median :-0.390280   Median :-0.48724   Median :-0.2109  
##  Mean   : 0.000000   Mean   : 0.00000   Mean   : 0.0000  
##  3rd Qu.: 0.007389   3rd Qu.: 0.04872   3rd Qu.: 1.0150  
##  Max.   : 9.924110   Max.   : 3.80047   Max.   : 2.4202  
##       chas              nox                rm               age         
##  Min.   :-0.2723   Min.   :-1.4644   Min.   :-3.8764   Min.   :-2.3331  
##  1st Qu.:-0.2723   1st Qu.:-0.9121   1st Qu.:-0.5681   1st Qu.:-0.8366  
##  Median :-0.2723   Median :-0.1441   Median :-0.1084   Median : 0.3171  
##  Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000  
##  3rd Qu.:-0.2723   3rd Qu.: 0.5981   3rd Qu.: 0.4823   3rd Qu.: 0.9059  
##  Max.   : 3.6648   Max.   : 2.7296   Max.   : 3.5515   Max.   : 1.1164  
##       dis               rad               tax             ptratio       
##  Min.   :-1.2658   Min.   :-0.9819   Min.   :-1.3127   Min.   :-2.7047  
##  1st Qu.:-0.8049   1st Qu.:-0.6373   1st Qu.:-0.7668   1st Qu.:-0.4876  
##  Median :-0.2790   Median :-0.5225   Median :-0.4642   Median : 0.2746  
##  Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000  
##  3rd Qu.: 0.6617   3rd Qu.: 1.6596   3rd Qu.: 1.5294   3rd Qu.: 0.8058  
##  Max.   : 3.9566   Max.   : 1.6596   Max.   : 1.7964   Max.   : 1.6372  
##      black             lstat              medv        
##  Min.   :-3.9033   Min.   :-1.5296   Min.   :-1.9063  
##  1st Qu.: 0.2049   1st Qu.:-0.7986   1st Qu.:-0.5989  
##  Median : 0.3808   Median :-0.1811   Median :-0.1449  
##  Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000  
##  3rd Qu.: 0.4332   3rd Qu.: 0.6024   3rd Qu.: 0.2683  
##  Max.   : 0.4406   Max.   : 3.5453   Max.   : 2.9865
# Class of the boston_scaled object
class(boston_scaled)
## [1] "matrix"
# Change the object to data frame
boston_scaled <- as.data.frame(boston_scaled)

Scaling of the dataset standardize the variables so that the mean is 0 and standard deviation is 1.

Creating a categorical variable of the crime rate

# Summary of the scaled crime rate
summary(boston_scaled$crim)
##      Min.   1st Qu.    Median      Mean   3rd Qu.      Max. 
## -0.419367 -0.410563 -0.390280  0.000000  0.007389  9.924110
# Create a quantile vector of crim and print it
bins <- quantile(boston_scaled$crim)
bins
##           0%          25%          50%          75%         100% 
## -0.419366929 -0.410563278 -0.390280295  0.007389247  9.924109610
# Create a categorical variable 'crime'
crime <- cut(boston_scaled$crim, breaks = bins, include.lowest = TRUE, label = c("low", "med_low", "med_high", "high"))

# Look at the table of the new factor crime
table(crime)
## crime
##      low  med_low med_high     high 
##      127      126      126      127
# Remove original crim from the dataset
boston_scaled <- dplyr::select(boston_scaled, -crim)

# Add the new categorical value to scaled data
boston_scaled <- data.frame(boston_scaled, crime)

Dividing the datset to a train and test sets

# Number of rows in the Boston dataset 
n <- nrow(boston_scaled)

# Choose randomly 80% of the rows
ind <- sample(n,  size = n * 0.8)

# Create train set
train <- boston_scaled[ind,]

# Create test set 
test <- boston_scaled[-ind,]

Linear discriminant analysis (LDA)

# Linear discriminant analysis
lda.fit <- lda(crime ~ ., data = train)

# Print the lda.fit object
lda.fit
## Call:
## lda(crime ~ ., data = train)
## 
## Prior probabilities of groups:
##       low   med_low  med_high      high 
## 0.2376238 0.2574257 0.2524752 0.2524752 
## 
## Group means:
##                   zn      indus         chas        nox           rm
## low       0.90649007 -0.9020745 -0.149294685 -0.8711616  0.384308674
## med_low  -0.08877141 -0.2744939 -0.083045403 -0.5604705 -0.156830901
## med_high -0.38467135  0.2387383  0.152260173  0.4273526 -0.001600564
## high     -0.48724019  1.0149946 -0.002135914  1.0192547 -0.454166225
##                 age        dis        rad        tax     ptratio
## low      -0.8836172  0.8892255 -0.6756133 -0.7543941 -0.40335515
## med_low  -0.3109464  0.3610218 -0.5456746 -0.4429328 -0.03897546
## med_high  0.4196039 -0.3895358 -0.4357864 -0.3093046 -0.34943789
## high      0.7795600 -0.8479649  1.6596029  1.5294129  0.80577843
##                black       lstat        medv
## low       0.37092984 -0.72649690  0.50633079
## med_low   0.31117058 -0.12155941 -0.04883629
## med_high  0.07705465  0.03076319  0.13458388
## high     -0.80492052  0.86835219 -0.64102258
## 
## Coefficients of linear discriminants:
##                  LD1          LD2         LD3
## zn       0.122775006  0.651207006 -0.98213652
## indus    0.081839720 -0.280515418  0.20703380
## chas     0.014865410 -0.005224141  0.10443371
## nox      0.227296558 -0.827318076 -1.31528961
## rm       0.052458662 -0.029663220 -0.08497085
## age      0.253118465 -0.352184210 -0.02424640
## dis     -0.101549847 -0.278106063  0.17861821
## rad      3.705305225  0.836121608 -0.32931968
## tax     -0.002452226  0.069321385  0.77794078
## ptratio  0.156282259  0.050665799 -0.20745654
## black   -0.125396447  0.026608391  0.10643251
## lstat    0.095205149 -0.122827397  0.27384024
## medv    -0.014018813 -0.343981230 -0.28716328
## 
## Proportion of trace:
##    LD1    LD2    LD3 
## 0.9586 0.0313 0.0102
# The function for lda biplot arrows
lda.arrows <- function(x, myscale = 1, arrow_heads = 0.1, color = "red", tex = 0.75, choices = c(1,2)){
  heads <- coef(x)
  arrows(x0 = 0, y0 = 0, 
         x1 = myscale * heads[,choices[1]], 
         y1 = myscale * heads[,choices[2]], col=color, length = arrow_heads)
  text(myscale * heads[,choices], labels = row.names(heads), 
       cex = tex, col=color, pos=3)
}

# Target classes as numeric
classes <- as.numeric(train$crime)

# Plot the lda results
plot(lda.fit, dimen = 2, col = classes, pch = classes)
lda.arrows(lda.fit, myscale = 3)

# Save the correct classes from test data
correct_classes <- test$crime

# Remove the crime variable from test data
test <- dplyr::select(test, -crime)

# Predict classes with test data
lda.pred <- predict(lda.fit, newdata = test)

# Cross tabulate the results
table(correct = correct_classes, predicted = lda.pred$class)
##           predicted
## correct    low med_low med_high high
##   low       20      10        1    0
##   med_low    2      17        3    0
##   med_high   0       7       15    2
##   high       0       0        1   24

LDA model works quite well. As assumed, the contiguous categories are harder to discriminate. However, the model does not mix extremity categories themselves at all.

Distances and K-means

# Reload the data and standardize it
data("Boston")
boston_scaled <- scale(Boston)

# Euclidean distance matrix
dist_eu <- dist(Boston)

# Look at the summary of the distances
summary(dist_eu)
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   1.119  85.624 170.539 226.315 371.950 626.047
# Manhattan distance matrix
dist_man <- dist(Boston, method = "manhattan")

# Look at the summary of the distances
summary(dist_man)
##     Min.  1st Qu.   Median     Mean  3rd Qu.     Max. 
##    2.016  149.145  279.505  342.899  509.707 1198.265
# K-means clustering
km <-kmeans(Boston, centers = 3)

# Plot the Boston dataset with clusters
pairs(Boston[6:10], col = km$cluster)

set.seed(123)

# Determine the number of clusters
k_max <- 10

# Calculate the total within sum of squares
twcss <- sapply(1:k_max, function(k){kmeans(Boston, k)$tot.withinss})

# Visualize the results
qplot(x = 1:k_max, y = twcss, geom = 'line')

As can be seen from the total WCSS picture, the value of total WCSS radically changes between 1 and 2,5. It indicates that the optimal number of clusters could be 2. Let’s run the algorithm again.

# K-means clustering
km <-kmeans(Boston, centers = 2)

# Plot the Boston dataset with clusters
pairs(Boston, col = km$cluster)

Now the colors red and black visualize the two clusters.

Dimensionality reduction techniques

Overview of the data

The dataset is a combination of two datasets; Human Development and Gender Inequality. After some “data wrangling”, the dataset looks like this. The wrangling steps can be seen from Rscript-file in my GitHub.

I decided to download the data from the link because somehow the mutation from string values to numeric did not saved in to the human data and I didn’t have time to figure out what was the problem.

human2 <- read.table("http://s3.amazonaws.com/assets.datacamp.com/production/course_2218/datasets/human2.txt", 
                  sep = ",", header = T)
str(human2) # 155 observations and 8 variables
## 'data.frame':    155 obs. of  8 variables:
##  $ Edu2.FM  : num  1.007 0.997 0.983 0.989 0.969 ...
##  $ Labo.FM  : num  0.891 0.819 0.825 0.884 0.829 ...
##  $ Edu.Exp  : num  17.5 20.2 15.8 18.7 17.9 16.5 18.6 16.5 15.9 19.2 ...
##  $ Life.Exp : num  81.6 82.4 83 80.2 81.6 80.9 80.9 79.1 82 81.8 ...
##  $ GNI      : int  64992 42261 56431 44025 45435 43919 39568 52947 42155 32689 ...
##  $ Mat.Mor  : int  4 6 6 5 6 7 9 28 11 8 ...
##  $ Ado.Birth: num  7.8 12.1 1.9 5.1 6.2 3.8 8.2 31 14.5 25.3 ...
##  $ Parli.F  : num  39.6 30.5 28.5 38 36.9 36.9 19.9 19.4 28.2 31.4 ...
summary(human2)
##     Edu2.FM          Labo.FM          Edu.Exp         Life.Exp    
##  Min.   :0.1717   Min.   :0.1857   Min.   : 5.40   Min.   :49.00  
##  1st Qu.:0.7264   1st Qu.:0.5984   1st Qu.:11.25   1st Qu.:66.30  
##  Median :0.9375   Median :0.7535   Median :13.50   Median :74.20  
##  Mean   :0.8529   Mean   :0.7074   Mean   :13.18   Mean   :71.65  
##  3rd Qu.:0.9968   3rd Qu.:0.8535   3rd Qu.:15.20   3rd Qu.:77.25  
##  Max.   :1.4967   Max.   :1.0380   Max.   :20.20   Max.   :83.50  
##       GNI            Mat.Mor         Ado.Birth         Parli.F     
##  Min.   :   581   Min.   :   1.0   Min.   :  0.60   Min.   : 0.00  
##  1st Qu.:  4198   1st Qu.:  11.5   1st Qu.: 12.65   1st Qu.:12.40  
##  Median : 12040   Median :  49.0   Median : 33.60   Median :19.30  
##  Mean   : 17628   Mean   : 149.1   Mean   : 47.16   Mean   :20.91  
##  3rd Qu.: 24512   3rd Qu.: 190.0   3rd Qu.: 71.95   3rd Qu.:27.95  
##  Max.   :123124   Max.   :1100.0   Max.   :204.80   Max.   :57.50

Let’s investigate the association between variables with a correlation matrix:

library("dplyr")
library("stringr")
library("MASS")
library("tidyr")
## 
## Attaching package: 'tidyr'
## The following object is masked from 'package:magrittr':
## 
##     extract
library("corrplot")
library("GGally")
ggpairs(human2) 

cor(human2) %>% corrplot()

Perform principal component analysis (PCA)

pca_human <- prcomp(human2) # PCA
biplot(pca_human, choices = 1:2, cex = c(0.8, 1), col = c("grey40", "deeppink2")) #Biplot
## Warning in arrows(0, 0, y[, 1L] * 0.8, y[, 2L] * 0.8, col = col[2L], length
## = arrow.len): zero-length arrow is of indeterminate angle and so skipped

## Warning in arrows(0, 0, y[, 1L] * 0.8, y[, 2L] * 0.8, col = col[2L], length
## = arrow.len): zero-length arrow is of indeterminate angle and so skipped

## Warning in arrows(0, 0, y[, 1L] * 0.8, y[, 2L] * 0.8, col = col[2L], length
## = arrow.len): zero-length arrow is of indeterminate angle and so skipped

## Warning in arrows(0, 0, y[, 1L] * 0.8, y[, 2L] * 0.8, col = col[2L], length
## = arrow.len): zero-length arrow is of indeterminate angle and so skipped

## Warning in arrows(0, 0, y[, 1L] * 0.8, y[, 2L] * 0.8, col = col[2L], length
## = arrow.len): zero-length arrow is of indeterminate angle and so skipped

s <- summary(pca_human)
s
## Importance of components:
##                              PC1      PC2   PC3   PC4   PC5   PC6    PC7
## Standard deviation     1.854e+04 185.5219 25.19 11.45 3.766 1.566 0.1912
## Proportion of Variance 9.999e-01   0.0001  0.00  0.00 0.000 0.000 0.0000
## Cumulative Proportion  9.999e-01   1.0000  1.00  1.00 1.000 1.000 1.0000
##                           PC8
## Standard deviation     0.1591
## Proportion of Variance 0.0000
## Cumulative Proportion  1.0000
pca_pr <- round(100*s$importance[2, ], digits = 1)
pca_pr
## PC1 PC2 PC3 PC4 PC5 PC6 PC7 PC8 
## 100   0   0   0   0   0   0   0

Not really working with unstandardized data. Let’s try after standardation:

human2_std <- scale(human2)
summary(human2_std)
##     Edu2.FM           Labo.FM           Edu.Exp           Life.Exp      
##  Min.   :-2.8189   Min.   :-2.6247   Min.   :-2.7378   Min.   :-2.7188  
##  1st Qu.:-0.5233   1st Qu.:-0.5484   1st Qu.:-0.6782   1st Qu.:-0.6425  
##  Median : 0.3503   Median : 0.2316   Median : 0.1140   Median : 0.3056  
##  Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000  
##  3rd Qu.: 0.5958   3rd Qu.: 0.7350   3rd Qu.: 0.7126   3rd Qu.: 0.6717  
##  Max.   : 2.6646   Max.   : 1.6632   Max.   : 2.4730   Max.   : 1.4218  
##       GNI             Mat.Mor          Ado.Birth          Parli.F       
##  Min.   :-0.9193   Min.   :-0.6992   Min.   :-1.1325   Min.   :-1.8203  
##  1st Qu.:-0.7243   1st Qu.:-0.6496   1st Qu.:-0.8394   1st Qu.:-0.7409  
##  Median :-0.3013   Median :-0.4726   Median :-0.3298   Median :-0.1403  
##  Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000  
##  3rd Qu.: 0.3712   3rd Qu.: 0.1932   3rd Qu.: 0.6030   3rd Qu.: 0.6127  
##  Max.   : 5.6890   Max.   : 4.4899   Max.   : 3.8344   Max.   : 3.1850
pca_human2 <- prcomp(human2_std) # PCA

s2 <- summary(pca_human2)
s2
## Importance of components:
##                           PC1    PC2     PC3     PC4     PC5     PC6
## Standard deviation     2.0708 1.1397 0.87505 0.77886 0.66196 0.53631
## Proportion of Variance 0.5361 0.1624 0.09571 0.07583 0.05477 0.03595
## Cumulative Proportion  0.5361 0.6984 0.79413 0.86996 0.92473 0.96069
##                            PC7     PC8
## Standard deviation     0.45900 0.32224
## Proportion of Variance 0.02634 0.01298
## Cumulative Proportion  0.98702 1.00000
pca_pr2 <- round(100*s2$importance[2, ], digits = 1)
pca_pr2
##  PC1  PC2  PC3  PC4  PC5  PC6  PC7  PC8 
## 53.6 16.2  9.6  7.6  5.5  3.6  2.6  1.3
pc_lab <- paste0(names(pca_pr2), " (", pca_pr2, "%)")
biplot(pca_human2, cex = c(0.8, 1), col = c("grey40", "deeppink2"), xlab = pc_lab[1], ylab = pc_lab[2]) # Biplot

Now the biplot looks much better.

Interpretations

This data cleary need standardazion to be analyzed (and especially interpret) with PCA. The unstandardized PCA accuse that the first principal component (PC) explains 100 % of the variance and so cannot separete the effect of component. After standardazion, however, the first PCA explain 53,6 % of the variance and the second one 16,2 %. …Sorry, I didn’t have time to finish the intepretation.

Tea dataset & Multiple Correspondence Analysis (MCA)

First, a quick overview of the dataset:

library("FactoMineR")
data("tea")
str(tea) 
## 'data.frame':    300 obs. of  36 variables:
##  $ breakfast       : Factor w/ 2 levels "breakfast","Not.breakfast": 1 1 2 2 1 2 1 2 1 1 ...
##  $ tea.time        : Factor w/ 2 levels "Not.tea time",..: 1 1 2 1 1 1 2 2 2 1 ...
##  $ evening         : Factor w/ 2 levels "evening","Not.evening": 2 2 1 2 1 2 2 1 2 1 ...
##  $ lunch           : Factor w/ 2 levels "lunch","Not.lunch": 2 2 2 2 2 2 2 2 2 2 ...
##  $ dinner          : Factor w/ 2 levels "dinner","Not.dinner": 2 2 1 1 2 1 2 2 2 2 ...
##  $ always          : Factor w/ 2 levels "always","Not.always": 2 2 2 2 1 2 2 2 2 2 ...
##  $ home            : Factor w/ 2 levels "home","Not.home": 1 1 1 1 1 1 1 1 1 1 ...
##  $ work            : Factor w/ 2 levels "Not.work","work": 1 1 2 1 1 1 1 1 1 1 ...
##  $ tearoom         : Factor w/ 2 levels "Not.tearoom",..: 1 1 1 1 1 1 1 1 1 2 ...
##  $ friends         : Factor w/ 2 levels "friends","Not.friends": 2 2 1 2 2 2 1 2 2 2 ...
##  $ resto           : Factor w/ 2 levels "Not.resto","resto": 1 1 2 1 1 1 1 1 1 1 ...
##  $ pub             : Factor w/ 2 levels "Not.pub","pub": 1 1 1 1 1 1 1 1 1 1 ...
##  $ Tea             : Factor w/ 3 levels "black","Earl Grey",..: 1 1 2 2 2 2 2 1 2 1 ...
##  $ How             : Factor w/ 4 levels "alone","lemon",..: 1 3 1 1 1 1 1 3 3 1 ...
##  $ sugar           : Factor w/ 2 levels "No.sugar","sugar": 2 1 1 2 1 1 1 1 1 1 ...
##  $ how             : Factor w/ 3 levels "tea bag","tea bag+unpackaged",..: 1 1 1 1 1 1 1 1 2 2 ...
##  $ where           : Factor w/ 3 levels "chain store",..: 1 1 1 1 1 1 1 1 2 2 ...
##  $ price           : Factor w/ 6 levels "p_branded","p_cheap",..: 4 6 6 6 6 3 6 6 5 5 ...
##  $ age             : int  39 45 47 23 48 21 37 36 40 37 ...
##  $ sex             : Factor w/ 2 levels "F","M": 2 1 1 2 2 2 2 1 2 2 ...
##  $ SPC             : Factor w/ 7 levels "employee","middle",..: 2 2 4 6 1 6 5 2 5 5 ...
##  $ Sport           : Factor w/ 2 levels "Not.sportsman",..: 2 2 2 1 2 2 2 2 2 1 ...
##  $ age_Q           : Factor w/ 5 levels "15-24","25-34",..: 3 4 4 1 4 1 3 3 3 3 ...
##  $ frequency       : Factor w/ 4 levels "1/day","1 to 2/week",..: 1 1 3 1 3 1 4 2 3 3 ...
##  $ escape.exoticism: Factor w/ 2 levels "escape-exoticism",..: 2 1 2 1 1 2 2 2 2 2 ...
##  $ spirituality    : Factor w/ 2 levels "Not.spirituality",..: 1 1 1 2 2 1 1 1 1 1 ...
##  $ healthy         : Factor w/ 2 levels "healthy","Not.healthy": 1 1 1 1 2 1 1 1 2 1 ...
##  $ diuretic        : Factor w/ 2 levels "diuretic","Not.diuretic": 2 1 1 2 1 2 2 2 2 1 ...
##  $ friendliness    : Factor w/ 2 levels "friendliness",..: 2 2 1 2 1 2 2 1 2 1 ...
##  $ iron.absorption : Factor w/ 2 levels "iron absorption",..: 2 2 2 2 2 2 2 2 2 2 ...
##  $ feminine        : Factor w/ 2 levels "feminine","Not.feminine": 2 2 2 2 2 2 2 1 2 2 ...
##  $ sophisticated   : Factor w/ 2 levels "Not.sophisticated",..: 1 1 1 2 1 1 1 2 2 1 ...
##  $ slimming        : Factor w/ 2 levels "No.slimming",..: 1 1 1 1 1 1 1 1 1 1 ...
##  $ exciting        : Factor w/ 2 levels "exciting","No.exciting": 2 1 2 2 2 2 2 2 2 2 ...
##  $ relaxing        : Factor w/ 2 levels "No.relaxing",..: 1 1 2 2 2 2 2 2 2 2 ...
##  $ effect.on.health: Factor w/ 2 levels "effect on health",..: 2 2 2 2 2 2 2 2 2 2 ...
dim(tea) # 300 observations, 36 variables
## [1] 300  36

Second, MCA (with only few variables):

keep_columns <- c("Tea", "How", "how", "sugar", "where", "lunch")
tea_time <- dplyr::select(tea, one_of(keep_columns))

gather(tea_time) %>% ggplot(aes(value)) + facet_wrap("key", scales = "free") + geom_bar() +
theme(axis.text.x = element_text(angle = 45, hjust = 1, size = 8))
## Warning: attributes are not identical across measure variables;
## they will be dropped

mca <- MCA(tea_time, graph = FALSE)
summary(mca)
## 
## Call:
## MCA(X = tea_time, graph = FALSE) 
## 
## 
## Eigenvalues
##                        Dim.1   Dim.2   Dim.3   Dim.4   Dim.5   Dim.6
## Variance               0.279   0.261   0.219   0.189   0.177   0.156
## % of var.             15.238  14.232  11.964  10.333   9.667   8.519
## Cumulative % of var.  15.238  29.471  41.435  51.768  61.434  69.953
##                        Dim.7   Dim.8   Dim.9  Dim.10  Dim.11
## Variance               0.144   0.141   0.117   0.087   0.062
## % of var.              7.841   7.705   6.392   4.724   3.385
## Cumulative % of var.  77.794  85.500  91.891  96.615 100.000
## 
## Individuals (the 10 first)
##                       Dim.1    ctr   cos2    Dim.2    ctr   cos2    Dim.3
## 1                  | -0.298  0.106  0.086 | -0.328  0.137  0.105 | -0.327
## 2                  | -0.237  0.067  0.036 | -0.136  0.024  0.012 | -0.695
## 3                  | -0.369  0.162  0.231 | -0.300  0.115  0.153 | -0.202
## 4                  | -0.530  0.335  0.460 | -0.318  0.129  0.166 |  0.211
## 5                  | -0.369  0.162  0.231 | -0.300  0.115  0.153 | -0.202
## 6                  | -0.369  0.162  0.231 | -0.300  0.115  0.153 | -0.202
## 7                  | -0.369  0.162  0.231 | -0.300  0.115  0.153 | -0.202
## 8                  | -0.237  0.067  0.036 | -0.136  0.024  0.012 | -0.695
## 9                  |  0.143  0.024  0.012 |  0.871  0.969  0.435 | -0.067
## 10                 |  0.476  0.271  0.140 |  0.687  0.604  0.291 | -0.650
##                       ctr   cos2  
## 1                   0.163  0.104 |
## 2                   0.735  0.314 |
## 3                   0.062  0.069 |
## 4                   0.068  0.073 |
## 5                   0.062  0.069 |
## 6                   0.062  0.069 |
## 7                   0.062  0.069 |
## 8                   0.735  0.314 |
## 9                   0.007  0.003 |
## 10                  0.643  0.261 |
## 
## Categories (the 10 first)
##                        Dim.1     ctr    cos2  v.test     Dim.2     ctr
## black              |   0.473   3.288   0.073   4.677 |   0.094   0.139
## Earl Grey          |  -0.264   2.680   0.126  -6.137 |   0.123   0.626
## green              |   0.486   1.547   0.029   2.952 |  -0.933   6.111
## alone              |  -0.018   0.012   0.001  -0.418 |  -0.262   2.841
## lemon              |   0.669   2.938   0.055   4.068 |   0.531   1.979
## milk               |  -0.337   1.420   0.030  -3.002 |   0.272   0.990
## other              |   0.288   0.148   0.003   0.876 |   1.820   6.347
## tea bag            |  -0.608  12.499   0.483 -12.023 |  -0.351   4.459
## tea bag+unpackaged |   0.350   2.289   0.056   4.088 |   1.024  20.968
## unpackaged         |   1.958  27.432   0.523  12.499 |  -1.015   7.898
##                       cos2  v.test     Dim.3     ctr    cos2  v.test  
## black                0.003   0.929 |  -1.081  21.888   0.382 -10.692 |
## Earl Grey            0.027   2.867 |   0.433   9.160   0.338  10.053 |
## green                0.107  -5.669 |  -0.108   0.098   0.001  -0.659 |
## alone                0.127  -6.164 |  -0.113   0.627   0.024  -2.655 |
## lemon                0.035   3.226 |   1.329  14.771   0.218   8.081 |
## milk                 0.020   2.422 |   0.013   0.003   0.000   0.116 |
## other                0.102   5.534 |  -2.524  14.526   0.197  -7.676 |
## tea bag              0.161  -6.941 |  -0.065   0.183   0.006  -1.287 |
## tea bag+unpackaged   0.478  11.956 |   0.019   0.009   0.000   0.226 |
## unpackaged           0.141  -6.482 |   0.257   0.602   0.009   1.640 |
## 
## Categorical variables (eta2)
##                      Dim.1 Dim.2 Dim.3  
## Tea                | 0.126 0.108 0.410 |
## How                | 0.076 0.190 0.394 |
## how                | 0.708 0.522 0.010 |
## sugar              | 0.065 0.001 0.336 |
## where              | 0.702 0.681 0.055 |
## lunch              | 0.000 0.064 0.111 |
plot(mca, invisible=c("ind"), habillage = "quali")

Sorry, I ran out of time and couldn’t intepret the results. :(

Analysis of longitudinal data

library(dplyr)
library(tidyr)

RATS data

First, two variables need to be factored again.

RATS <- read.table("https://raw.githubusercontent.com/KimmoVehkalahti/MABS/master/Examples/data/rats.txt", header = TRUE, sep = '\t')

RATS$ID <- factor(RATS$ID)
RATS$Group <- factor(RATS$Group)
glimpse(RATS)
## Observations: 16
## Variables: 13
## $ ID    <fct> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16
## $ Group <fct> 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3
## $ WD1   <int> 240, 225, 245, 260, 255, 260, 275, 245, 410, 405, 445, 5...
## $ WD8   <int> 250, 230, 250, 255, 260, 265, 275, 255, 415, 420, 445, 5...
## $ WD15  <int> 255, 230, 250, 255, 255, 270, 260, 260, 425, 430, 450, 5...
## $ WD22  <int> 260, 232, 255, 265, 270, 275, 270, 268, 428, 440, 452, 5...
## $ WD29  <int> 262, 240, 262, 265, 270, 275, 273, 270, 438, 448, 455, 5...
## $ WD36  <int> 258, 240, 265, 268, 273, 277, 274, 265, 443, 460, 455, 5...
## $ WD43  <int> 266, 243, 267, 270, 274, 278, 276, 265, 442, 458, 451, 5...
## $ WD44  <int> 266, 244, 267, 272, 273, 278, 271, 267, 446, 464, 450, 5...
## $ WD50  <int> 265, 238, 264, 274, 276, 284, 282, 273, 456, 475, 462, 6...
## $ WD57  <int> 272, 247, 268, 273, 278, 279, 281, 274, 468, 484, 466, 6...
## $ WD64  <int> 278, 245, 269, 275, 280, 281, 284, 278, 478, 496, 472, 6...
RATSL <- RATS %>%
  gather(key = WD, value = Weight, -ID, -Group) %>%
  mutate(Time = as.integer(substr(WD,3,4)))
glimpse(RATSL)
## Observations: 176
## Variables: 5
## $ ID     <fct> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, ...
## $ Group  <fct> 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 1, 1, 1...
## $ WD     <chr> "WD1", "WD1", "WD1", "WD1", "WD1", "WD1", "WD1", "WD1",...
## $ Weight <int> 240, 225, 245, 260, 255, 260, 275, 245, 410, 405, 445, ...
## $ Time   <int> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 8, 8...

Let’s begin with plotting the data to identify the observations in each group (but ignoring the longitudinal nature of the data at this point).

library(ggplot2)

ggplot(RATSL, aes(x = Time, y = Weight, group = ID)) +
  geom_line(aes(linetype = Group)) +
  scale_x_continuous(name = "Time (days)", breaks = seq(0, 60, 10)) + 
  scale_y_continuous(name = "Weight (grams)") + 
  theme(legend.position = "top")

As we can see from the plot, group 1 cleary differs from two other groups. Rats in group 1 weigh less and their weight does not rise during the days compared to two other groups.

Next, we can build a linear model to examine does group and time predict weight as the plot suggest. We cannot use basic linear regression model because it assumes that the observations are independent and in this case they are not as we examine the weights of same rats. Let’s fit random intercept model to study if group and time predict the rat’s weight.

library(lme4)
## Loading required package: Matrix
## 
## Attaching package: 'Matrix'
## The following object is masked from 'package:tidyr':
## 
##     expand
RATS_ref <- lmer(Weight ~ Time + Group + (1 | ID), data = RATSL, REML = FALSE)
RATS_ref
## Linear mixed model fit by maximum likelihood  ['lmerMod']
## Formula: Weight ~ Time + Group + (1 | ID)
##    Data: RATSL
##       AIC       BIC    logLik  deviance  df.resid 
## 1333.1639 1352.1868 -660.5819 1321.1639       170 
## Random effects:
##  Groups   Name        Std.Dev.
##  ID       (Intercept) 32.953  
##  Residual              8.151  
## Number of obs: 176, groups:  ID, 16
## Fixed Effects:
## (Intercept)         Time       Group2       Group3  
##    244.0689       0.5857     220.9886     262.0795

After modelling we note that standard deviation is quite big suggesting large variablility between rats. Let’s take one step foward and see how the random slope model works.

Fitting a random intercept and random slope model allows the linear regression fits for each individual to differ in intercept but also in slope. This way it is possible to account for the individual differences in the rats’ growth profiles, but also the effect of time.

RATS_ref1 <- lmer(Weight ~ Time + Group + (Time | ID), data = RATSL, REML = FALSE)
RATS_ref1
## Linear mixed model fit by maximum likelihood  ['lmerMod']
## Formula: Weight ~ Time + Group + (Time | ID)
##    Data: RATSL
##       AIC       BIC    logLik  deviance  df.resid 
## 1194.2189 1219.5828 -589.1095 1178.2189       168 
## Random effects:
##  Groups   Name        Std.Dev. Corr 
##  ID       (Intercept) 33.7718       
##           Time         0.3349  -0.22
##  Residual              4.4436       
## Number of obs: 176, groups:  ID, 16
## Fixed Effects:
## (Intercept)         Time       Group2       Group3  
##    246.4573       0.5857     214.5874     258.9273

The two models seem not to be very different. We can also perform an ANOVA test to compare the models.

anova(RATS_ref1, RATS_ref)
## Data: RATSL
## Models:
## RATS_ref: Weight ~ Time + Group + (1 | ID)
## RATS_ref1: Weight ~ Time + Group + (Time | ID)
##           Df    AIC    BIC  logLik deviance  Chisq Chi Df Pr(>Chisq)    
## RATS_ref   6 1333.2 1352.2 -660.58   1321.2                             
## RATS_ref1  8 1194.2 1219.6 -589.11   1178.2 142.94      2  < 2.2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

The interpretation of the test can be done by looking the values of chi-squared statistics and p-value. Lower value address better fit against the comparison model: the random slope model seems to fit better.

Finally, we can fit a random intercept and slope model that allows for a group × time interaction.

RATS_ref2 <- lmer(Weight ~ Time * Group + (Time | ID), data = RATSL, REML = FALSE)
summary(RATS_ref2)
## Linear mixed model fit by maximum likelihood  ['lmerMod']
## Formula: Weight ~ Time * Group + (Time | ID)
##    Data: RATSL
## 
##      AIC      BIC   logLik deviance df.resid 
##   1185.9   1217.6   -582.9   1165.9      166 
## 
## Scaled residuals: 
##     Min      1Q  Median      3Q     Max 
## -3.2669 -0.4249  0.0726  0.6034  2.7513 
## 
## Random effects:
##  Groups   Name        Variance  Std.Dev. Corr 
##  ID       (Intercept) 1.107e+03 33.2763       
##           Time        4.925e-02  0.2219  -0.15
##  Residual             1.975e+01  4.4436       
## Number of obs: 176, groups:  ID, 16
## 
## Fixed effects:
##              Estimate Std. Error t value
## (Intercept) 251.65165   11.80279  21.321
## Time          0.35964    0.08215   4.378
## Group2      200.66549   20.44303   9.816
## Group3      252.07168   20.44303  12.330
## Time:Group2   0.60584    0.14229   4.258
## Time:Group3   0.29834    0.14229   2.097
## 
## Correlation of Fixed Effects:
##             (Intr) Time   Group2 Group3 Tm:Gr2
## Time        -0.160                            
## Group2      -0.577  0.092                     
## Group3      -0.577  0.092  0.333              
## Time:Group2  0.092 -0.577 -0.160 -0.053       
## Time:Group3  0.092 -0.577 -0.053 -0.160  0.333
anova(RATS_ref2, RATS_ref1)
## Data: RATSL
## Models:
## RATS_ref1: Weight ~ Time + Group + (Time | ID)
## RATS_ref2: Weight ~ Time * Group + (Time | ID)
##           Df    AIC    BIC  logLik deviance  Chisq Chi Df Pr(>Chisq)   
## RATS_ref1  8 1194.2 1219.6 -589.11   1178.2                            
## RATS_ref2 10 1185.9 1217.6 -582.93   1165.9 12.361      2    0.00207 **
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

The model with interaction fits even better than the random slope model without it. We should also check the plot with observed values.

Fitted <- fitted(RATS_ref2)
RATSL <- RATSL %>%
  mutate(Fitted)

ggplot(RATSL, aes(x = Time, y = Fitted, group = ID)) +
  geom_line(aes(linetype = Group)) +
  scale_x_continuous(name = "Time (days)", breaks = seq(0, 60, 20)) +
  scale_y_continuous(name = "Fitted weight (grams)") +
  theme(legend.position = "top")

BPRS data

First, two variables need to be factored again.

BPRS <- read.table("https://raw.githubusercontent.com/KimmoVehkalahti/MABS/master/Examples/data/BPRS.txt", sep  =" ", header = T)

BPRS$treatment <- factor(BPRS$treatment)
BPRS$subject <- factor(BPRS$subject)
glimpse(BPRS)
## Observations: 40
## Variables: 11
## $ treatment <fct> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...
## $ subject   <fct> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 1...
## $ week0     <int> 42, 58, 54, 55, 72, 48, 71, 30, 41, 57, 30, 55, 36, ...
## $ week1     <int> 36, 68, 55, 77, 75, 43, 61, 36, 43, 51, 34, 52, 32, ...
## $ week2     <int> 36, 61, 41, 49, 72, 41, 47, 38, 39, 51, 34, 49, 36, ...
## $ week3     <int> 43, 55, 38, 54, 65, 38, 30, 38, 35, 55, 41, 54, 31, ...
## $ week4     <int> 41, 43, 43, 56, 50, 36, 27, 31, 28, 53, 36, 48, 25, ...
## $ week5     <int> 40, 34, 28, 50, 39, 29, 40, 26, 22, 43, 36, 43, 25, ...
## $ week6     <int> 38, 28, 29, 47, 32, 33, 30, 26, 20, 43, 38, 37, 21, ...
## $ week7     <int> 47, 28, 25, 42, 38, 27, 31, 25, 23, 39, 36, 36, 19, ...
## $ week8     <int> 51, 28, 24, 46, 32, 25, 31, 24, 21, 32, 36, 31, 22, ...
BPRSL <-  BPRS %>% gather(key = weeks, value = bprs, -treatment, -subject)
BPRSL <-  BPRSL %>% mutate(week = as.integer(substr(weeks,5,5)))
glimpse(BPRSL)
## Observations: 360
## Variables: 5
## $ treatment <fct> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...
## $ subject   <fct> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 1...
## $ weeks     <chr> "week0", "week0", "week0", "week0", "week0", "week0"...
## $ bprs      <int> 42, 58, 54, 55, 72, 48, 71, 30, 41, 57, 30, 55, 36, ...
## $ week      <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...

Let’s begin by plotting the BPRS values for all 40 men, differentiating between the treatment groups into which the men have been randomized.

ggplot(BPRSL, aes(x = week, y = bprs, linetype = subject)) +
  geom_line() +
  scale_linetype_manual(values = rep(1:10, times=4)) +
  facet_grid(. ~ treatment, labeller = label_both) +
  theme(legend.position = "none") + 
  scale_y_continuous(limits = c(min(BPRSL$bprs), max(BPRSL$bprs)))

At this point the plot mostly shows that even though there is a lot of variability between individuals, there seems to be less psychotic symptoms in group 1 after 8 weeks.

Next we should do some analysis to examine if this hypothesis is correct. First we can standardize the values of BPRS.

BPRSL <- BPRSL %>%
  group_by(week) %>%
  mutate(stdbprs = (bprs - mean(bprs)) / sd(bprs)) %>%
  ungroup()
glimpse(BPRSL)
## Observations: 360
## Variables: 6
## $ treatment <fct> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...
## $ subject   <fct> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 1...
## $ weeks     <chr> "week0", "week0", "week0", "week0", "week0", "week0"...
## $ bprs      <int> 42, 58, 54, 55, 72, 48, 71, 30, 41, 57, 30, 55, 36, ...
## $ week      <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...
## $ stdbprs   <dbl> -0.4245908, 0.7076513, 0.4245908, 0.4953559, 1.69836...

Let’s plot the data again, now with the standardized values.

ggplot(BPRSL, aes(x = week, y = stdbprs, linetype = subject)) +
  geom_line() +
  scale_linetype_manual(values = rep(1:10, times=4)) +
  facet_grid(. ~ treatment, labeller = label_both) +
  scale_y_continuous(name = "standardized bprs")

After standardizing the values, the plot shows that the variability is still large but now in group 1 psychotic symptoms do not seem to reduce.

Next we can try produce a graph showing average profiles for each treatment group along with some indication of the variation of the observations at each time point.

n <- BPRSL$week %>% unique() %>% length()
BPRSS <- BPRSL %>%
  group_by(treatment, week) %>%
  summarise( mean = mean(bprs), se = sd(bprs) / sqrt(n) ) %>%
  ungroup()
glimpse(BPRSS)
## Observations: 18
## Variables: 4
## $ treatment <fct> 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2
## $ week      <int> 0, 1, 2, 3, 4, 5, 6, 7, 8, 0, 1, 2, 3, 4, 5, 6, 7, 8
## $ mean      <dbl> 47.00, 46.80, 43.55, 40.90, 36.60, 32.70, 29.70, 29....
## $ se        <dbl> 4.534468, 5.173708, 4.003617, 3.744626, 3.259534, 2....
# Plot the mean profiles
ggplot(BPRSS, aes(x = week, y = mean, linetype = treatment, shape = treatment)) +
  geom_line() +
  scale_linetype_manual(values = c(1,2)) +
  geom_point(size=3) +
  scale_shape_manual(values = c(1,2)) +
  geom_errorbar(aes(ymin=mean-se, ymax=mean+se, linetype="1"), width=0.3) +
  theme(legend.position = c(0.8,0.8)) +
  scale_y_continuous(name = "mean(bprs) +/- se(bprs)")

Okay, the two groups seems to overlap quite much. Let’s find out if there are an outlier which could explain this situation.

BPRSL8S <- BPRSL %>%
  filter(week > 0) %>%
  group_by(treatment, subject) %>%
  summarise( mean=mean(bprs) ) %>%
  ungroup()
glimpse(BPRSL8S)
## Observations: 40
## Variables: 3
## $ treatment <fct> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...
## $ subject   <fct> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 1...
## $ mean      <dbl> 41.500, 43.125, 35.375, 52.625, 50.375, 34.000, 37.1...
# Draw a boxplot of the mean versus treatment
ggplot(BPRSL8S, aes(x = treatment, y = mean)) +
  geom_boxplot() +
  stat_summary(fun.y = "mean", geom = "point", shape=23, size=4, fill = "white") +
  scale_y_continuous(name = "mean(bprs), weeks 1-8")

Hahaa, there is one outlier in group 2. Let’s filter that one out and see the situation once again.

BPRSL8S1 <- BPRSL8S %>% filter(mean < 70)
glimpse(BPRSL8S1)
## Observations: 39
## Variables: 3
## $ treatment <fct> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...
## $ subject   <fct> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 1...
## $ mean      <dbl> 41.500, 43.125, 35.375, 52.625, 50.375, 34.000, 37.1...
ggplot(BPRSL8S1, aes(x = treatment, y = mean)) +
  geom_boxplot() +
  stat_summary(fun.y = "mean", geom = "point", shape=23, size=4, fill = "white") +
  scale_y_continuous(name = "mean(bprs), weeks 1-8")

Now it looks like that there are less psychotic symptoms in group 2. Next we should do some analysis to be more sure about it.

t.test(mean ~ treatment, data = BPRSL8S1, var.equal = TRUE)
## 
##  Two Sample t-test
## 
## data:  mean by treatment
## t = 0.52095, df = 37, p-value = 0.6055
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -4.232480  7.162085
## sample estimates:
## mean in group 1 mean in group 2 
##        36.16875        34.70395
BPRSL8S2 <- BPRSL8S %>% mutate(baseline = BPRS$week0)

fit <- lm(mean ~ baseline + treatment, data = BPRSL8S2)

anova(fit)
## Analysis of Variance Table
## 
## Response: mean
##           Df  Sum Sq Mean Sq F value    Pr(>F)    
## baseline   1 1868.07 1868.07 30.1437 3.077e-06 ***
## treatment  1    3.45    3.45  0.0557    0.8148    
## Residuals 37 2292.97   61.97                      
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Yep, the t-test confirms that the BPRS score is lower in group 2 after 8 weeks.